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Theorem pellexlem3 30371
Description: Lemma for pellex 30375. To each good rational approximation of  ( sqr `  D
), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
Assertion
Ref Expression
pellexlem3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Distinct variable group:    x, D, y, z

Proof of Theorem pellexlem3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 10538 . . . 4  |-  NN  e.  _V
21, 1xpex 6711 . . 3  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 5072 . . 3  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  C_  ( NN  X.  NN )
42, 3ssexi 4592 . 2  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V
5 simprl 755 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  a  e.  QQ )
6 simprrl 763 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  0  <  a )
7 qgt0numnn 14139 . . . . . . . 8  |-  ( ( a  e.  QQ  /\  0  <  a )  -> 
(numer `  a )  e.  NN )
85, 6, 7syl2anc 661 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (numer `  a )  e.  NN )
9 qdencl 14129 . . . . . . . 8  |-  ( a  e.  QQ  ->  (denom `  a )  e.  NN )
105, 9syl 16 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (denom `  a )  e.  NN )
118, 10jca 532 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) )
12 simpll 753 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  D  e.  NN )
13 simplr 754 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  -.  ( sqr `  D )  e.  QQ )
14 pellexlem1 30369 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  -.  ( sqr `  D )  e.  QQ )  ->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
1512, 8, 10, 13, 14syl31anc 1231 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
16 simprrr 764 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
17 qeqnumdivden 14134 . . . . . . . . . . . 12  |-  ( a  e.  QQ  ->  a  =  ( (numer `  a )  /  (denom `  a ) ) )
1817oveq1d 6297 . . . . . . . . . . 11  |-  ( a  e.  QQ  ->  (
a  -  ( sqr `  D ) )  =  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )
1918fveq2d 5868 . . . . . . . . . 10  |-  ( a  e.  QQ  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  =  ( abs `  ( ( (numer `  a )  /  (denom `  a )
)  -  ( sqr `  D ) ) ) )
2019breq1d 4457 . . . . . . . . 9  |-  ( a  e.  QQ  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
215, 20syl 16 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
2216, 21mpbid 210 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
23 pellexlem2 30370 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )  ->  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) )
2412, 8, 10, 22, 23syl31anc 1231 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) )
2511, 15, 24jca32 535 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
)  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
2625ex 434 . . . 4  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  QQ  /\  (
0  <  a  /\  ( abs `  ( a  -  ( sqr `  D
) ) )  < 
( (denom `  a
) ^ -u 2
) ) )  -> 
( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
27 breq2 4451 . . . . . 6  |-  ( x  =  a  ->  (
0  <  x  <->  0  <  a ) )
28 oveq1 6289 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  ( sqr `  D ) )  =  ( a  -  ( sqr `  D ) ) )
2928fveq2d 5868 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  ( sqr `  D ) ) )  =  ( abs `  ( a  -  ( sqr `  D
) ) ) )
30 fveq2 5864 . . . . . . . 8  |-  ( x  =  a  ->  (denom `  x )  =  (denom `  a ) )
3130oveq1d 6297 . . . . . . 7  |-  ( x  =  a  ->  (
(denom `  x ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
3229, 31breq12d 4460 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
)  <->  ( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
) ) )
3327, 32anbi12d 710 . . . . 5  |-  ( x  =  a  ->  (
( 0  <  x  /\  ( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
3433elrab 3261 . . . 4  |-  ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  <->  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
35 fvex 5874 . . . . 5  |-  (numer `  a )  e.  _V
36 fvex 5874 . . . . 5  |-  (denom `  a )  e.  _V
37 eleq1 2539 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( y  e.  NN  <->  (numer `  a )  e.  NN ) )
3837anbi1d 704 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
y  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  z  e.  NN ) ) )
39 oveq1 6289 . . . . . . . . 9  |-  ( y  =  (numer `  a
)  ->  ( y ^ 2 )  =  ( (numer `  a
) ^ 2 ) )
4039oveq1d 6297 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( (
y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )
4140neeq1d 2744 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  <->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0
) )
4240fveq2d 5868 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) ) )
4342breq1d 4457 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( ( abs `  ( ( y ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) )  < 
( 1  +  ( 2  x.  ( sqr `  D ) ) )  <-> 
( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
4441, 43anbi12d 710 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
4538, 44anbi12d 710 . . . . 5  |-  ( y  =  (numer `  a
)  ->  ( (
( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
46 eleq1 2539 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( z  e.  NN  <->  (denom `  a )  e.  NN ) )
4746anbi2d 703 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a )  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) ) )
48 oveq1 6289 . . . . . . . . . 10  |-  ( z  =  (denom `  a
)  ->  ( z ^ 2 )  =  ( (denom `  a
) ^ 2 ) )
4948oveq2d 6298 . . . . . . . . 9  |-  ( z  =  (denom `  a
)  ->  ( D  x.  ( z ^ 2 ) )  =  ( D  x.  ( (denom `  a ) ^ 2 ) ) )
5049oveq2d 6298 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =  ( ( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )
5150neeq1d 2744 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  <->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 ) )
5250fveq2d 5868 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) ) )
5352breq1d 4457 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) )  <->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
5451, 53anbi12d 710 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
5547, 54anbi12d 710 . . . . 5  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
)  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
5635, 36, 45, 55opelopab 4769 . . . 4  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  <->  ( (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
(denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
5726, 34, 563imtr4g 270 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ->  <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
58 ssrab2 3585 . . . . . 6  |-  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  C_  QQ
59 simprl 755 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6058, 59sseldi 3502 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  QQ )
61 simprr 756 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6258, 61sseldi 3502 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  QQ )
6335, 36opth 4721 . . . . . . 7  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b
) ,  (denom `  b ) >.  <->  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )
64 simprl 755 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (numer `  a
)  =  (numer `  b ) )
65 simprr 756 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (denom `  a
)  =  (denom `  b ) )
6664, 65oveq12d 6300 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  ( (numer `  a )  /  (denom `  a ) )  =  ( (numer `  b
)  /  (denom `  b ) ) )
67 simpll 753 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  e.  QQ )
6867, 17syl 16 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  ( (numer `  a )  /  (denom `  a )
) )
69 simplr 754 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  e.  QQ )
70 qeqnumdivden 14134 . . . . . . . . . 10  |-  ( b  e.  QQ  ->  b  =  ( (numer `  b )  /  (denom `  b ) ) )
7169, 70syl 16 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  =  ( (numer `  b )  /  (denom `  b )
) )
7266, 68, 713eqtr4d 2518 . . . . . . . 8  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  b )
7372ex 434 . . . . . . 7  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) )  -> 
a  =  b ) )
7463, 73syl5bi 217 . . . . . 6  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  ->  a  =  b ) )
75 fveq2 5864 . . . . . . 7  |-  ( a  =  b  ->  (numer `  a )  =  (numer `  b ) )
76 fveq2 5864 . . . . . . 7  |-  ( a  =  b  ->  (denom `  a )  =  (denom `  b ) )
7775, 76opeq12d 4221 . . . . . 6  |-  ( a  =  b  ->  <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >. )
7874, 77impbid1 203 . . . . 5  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
7960, 62, 78syl2anc 661 . . . 4  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
8079ex 434 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  /\  b  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) } )  ->  ( <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) ) )
8157, 80dom2d 7553 . 2  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V  ->  { x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
824, 81mpi 17 1  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113   <.cop 4033   class class class wbr 4447   {copab 4504    X. cxp 4997   ` cfv 5586  (class class class)co 6282    ~<_ cdom 7511   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    - cmin 9801   -ucneg 9802    / cdiv 10202   NNcn 10532   2c2 10581   QQcq 11178   ^cexp 12130   sqrcsqrt 13025   abscabs 13026  numercnumer 14121  denomcdenom 14122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-dvds 13844  df-gcd 14000  df-numer 14123  df-denom 14124
This theorem is referenced by:  pellexlem4  30372
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